
// Make newform 1620.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1620_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1620_2_a_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1620_2_a_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1620_a();"
function MakeCharacter_1620_a()
    N := 1620;
    order := 1;
    char_gens := [811, 1541, 1297];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_1620_a_Hecke(Kf)
    return MakeCharacter_1620_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0], [0], [-1], [2], [3], [-4], [6], [-7], [6], [3], [5], [-4], [3], [8], [0], [-6], [-3], [14], [2], [15], [-10], [8], [0], [15], [8], [-3], [2], [-6], [11], [18], [2], [-15], [12], [-1], [6], [5], [-4], [-22], [-18], [18], [-15], [23], [-21], [-16], [24], [-16], [5], [-28], [-18], [14], [-24], [0], [-7], [12], [-18], [6], [-15], [8], [-28], [30], [-22], [24], [20], [-15], [-16], [-12], [17], [-28], [18], [-7], [-24], [27], [-22], [-22], [32], [-24], [6], [2], [-6], [-10], [-36], [-7], [-3], [2], [-19], [0], [9], [8], [-15], [14], [-6], [27], [-16], [-39], [5], [-36], [30], [42], [-28], [-13], [32], [24], [36], [-3], [-37], [26], [-12], [-30], [-39], [17], [8], [8], [-24], [-4], [5], [-33], [-40], [18], [24], [24], [-7], [-10], [6], [36], [44], [3], [26], [9], [44], [14], [-25], [54], [8], [-28], [9], [41], [36], [-34], [0], [15], [-7], [15], [14], [-12], [-25], [-9], [14], [0], [-55], [6], [-28], [-33], [2], [48], [26], [45], [11], [9], [-58], [6], [-54], [0], [2], [-45], [24], [42], [-19], [26], [53], [-42], [3], [23], [3], [38], [-40], [15], [-28], [9], [-16], [35], [-52], [24], [14], [18], [-24], [-21], [-10], [2], [23], [24], [2], [-36], [32], [-21], [-12], [6], [-31], [8], [6], [24], [27], [-37], [50], [41], [-24], [-48], [-7], [-60], [-66], [-28], [-22], [15], [-4], [-12], [27], [5], [14], [-27], [-42], [6], [5], [23], [-69], [20], [18], [-73], [-18], [12], [-52], [-24], [62], [11], [-40], [-6], [-16], [-42], [59], [-18], [-36], [21], [-30], [44], [56], [-10], [-12], [21], [-52], [12], [-31], [72], [-76], [-63], [60], [23], [-36], [60], [71], [-46], [-54], [-70], [44], [36], [-79], [-4], [-24], [-1], [42], [45], [-40], [-18], [35], [68], [44], [-25], [-22], [8], [66], [-19], [-70], [-48], [-24], [-40], [-78], [-22], [2], [-81], [20], [-12], [-31], [18], [27], [-24], [-30], [27], [62], [69], [-19], [-66], [-27], [-10], [50], [18], [47], [-6], [-19], [2], [30], [-7], [-3], [-52], [-6], [-9], [45], [-40], [42], [-79], [75], [51], [44], [45], [32], [14], [-33], [74], [60], [-31], [-79], [62], [36], [30], [41], [24], [-25], [18], [-61], [12], [35], [30], [5], [20], [-64], [66], [9], [-88], [42], [12], [29], [44], [-75], [12], [-1], [-88], [78], [62], [65], [66], [-27], [-3], [42], [84], [-64], [21], [-18], [-3], [-16], [56], [60], [-16], [65], [75], [-4], [-6], [69], [-67], [2], [-45], [99], [20], [-3], [86], [66], [42], [-76], [-30], [95], [-66], [-28], [-46], [-70], [18], [-10], [-12], [-12], [20], [-63], [-16], [5], [-18], [68], [3], [23], [60], [-22], [-90], [45], [-16], [-46], [-78], [-16], [-75], [-46], [-42], [30], [23], [8], [-30], [3], [26], [-78], [-6], [30], [-52], [-84], [-9], [-52], [24], [-60], [54], [-25], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1620_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1620_2_a_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1620_2_a_c(:prec:=1)
    chi := MakeCharacter_1620_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1620_2_a_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1620_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1620_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![-3, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;